Question

simplify the expression below.
\\(\frac{36x^{9}y^{8}}{45x^{3}y^{5}}\\)

Explanation:

Step1: Simplify the coefficients

First, simplify the fraction of the coefficients \(\frac{36}{45}\). We can find the greatest common divisor (GCD) of 36 and 45, which is 9. Divide both the numerator and the denominator by 9: \(\frac{36\div9}{45\div9}=\frac{4}{5}\).

Step2: Simplify the \(x\)-terms

For the \(x\)-terms, we use the quotient rule of exponents, which states that \(\frac{x^m}{x^n}=x^{m - n}\) (where \(x
eq0\) and \(m,n\) are real numbers). Here, \(m = 9\) and \(n = 3\), so \(\frac{x^9}{x^3}=x^{9 - 3}=x^6\).

Step3: Simplify the \(y\)-terms

Similarly, for the \(y\)-terms, using the quotient rule of exponents with \(m = 8\) and \(n = 5\), we have \(\frac{y^8}{y^5}=y^{8 - 5}=y^3\).

Step4: Combine the simplified terms

Now, multiply the simplified coefficient, \(x\)-term, and \(y\)-term together: \(\frac{4}{5}\times x^6\times y^3=\frac{4}{5}x^6y^3\).

Answer:

\(\frac{4}{5}x^6y^3\)